Triangle Inequality Theorem

Last updated on 17 November 2025

Table of contents

What is Triangle Inequality Theorem?
Solved Examples on Triangle Inequality Theorem
FAQs on Triangle Inequality Theorem

The Triangle Inequality Theorem is a fundamental concept in geometry that deals with the relationship between the lengths of the sides of a triangle. This theorem establishes a crucial condition, that three sides of a triangle must satisfy to form a valid triangle. It is a simple yet powerful principle that has significant implications in various fields of mathematics, including geometry. Understanding this theorem is essential for grasping the fundamental properties and relationships within geometric shapes.

What is Triangle Inequality Theorem?

In any triangle, the sum of the lengths of any two sides is always greater than the length of the third side.

Consider the triangle shown above.

a

b

, and

c

are the lengths of the sides of the triangle.
The triangle inequality theorem states that:

\begin{matrix} a + b > c \\ a + c > b \\ b + c > a \end{matrix}

This rule applies to all types of triangles, whether they are equilateral, isosceles, or scalene. It serves as a crucial criterion for determining whether a given set of side lengths can indeed form a valid triangle. If the inequality is not satisfied for any of the three possible combinations, it indicates that the sides cannot form a triangle.

This theorem is instrumental in geometric constructions. When constructing triangles, adherence to the Triangle Inequality Theorem ensures that the sides are connected in a way that forms a valid shape. Violating this theorem during construction results in a figure that fails to meet the criteria of a triangle.

Solved Examples on Triangle Inequality Theorem

Example 1: Can a triangle have side lengths as

2 cm

3 cm

and

4 cm

Given:

a = 2 cm, b = 3 cm, c = 4 cm

Solution: Given side lengths should satisfy the Triangle Inequality Theorem to form a valid triangle.
Let's check if the sum of the two sides is greater than the third side.

\begin{array}{l} a + b > c \\ 2 + 3 > 4 => 5 > 4 => TRUE \\ a + c > b \\ 2 + 4 > 3 => 6 > 3 => TRUE \\ b + c > a \\ 3 + 4 > 2 => 7 > 2 => TRUE \end{array}

All the three conditions are satisfied, therefore a triangle can have side lengths as

2 cm

3 cm

and

4 cm

Example 2: If

6 inches

2 inches

and

4 inches

are the measures of three line segments. Can these line segments be used to draw a triangle?

Given:

a = 6 inches, b = 2 inches, c = 4 inches

Solution: The triangle formed by the given side lengths must satisfy the triangle inequality theorem.
Hence, let's check if the sum of the two sides is greater than the third side.

\begin{array}{l} a + b > c \\ 6 + 2 > 4 => 8 > 4 => TRUE \\ a + c > b \\ 6 + 4 > 2 => 10 > 2 => TRUE \\ b + c > a \\ 2 + 4 > 6 => 6 ≯ 6 => FALSE \end{array}

The sides of the triangle do not satisfy the triangle inequality theorem. So, the given line segments can not form a triangle.

FAQs on Triangle Inequality Theorem

What is the Triangle Inequality Theorem?
The Triangle Inequality Theorem states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This can be expressed as:
$a + b > c$ , $a + c > b$ , $b + c > a$
where $a$ , $b$ and $c$ are the lengths of the sides of the triangle.
Why is the Triangle Inequality Theorem important?
It is fundamental in geometry as it provides a necessary condition for three lengths to form a triangle. It helps in determining whether a given set of lengths can create a triangle.
Does the Triangle Inequality Theorem apply to all triangles?
Yes, the theorem applies to all types of triangles, including equilateral, isosceles, and scalene triangles.
What happens if the Triangle Inequality Theorem is not satisfied?
If the inequality conditions are not met (i.e., if the sum of the lengths of any two sides is not greater than the length of the third side), then those lengths cannot form a triangle.
Can the Triangle Inequality be an equality?
Yes, if the sum of the lengths of two sides equals the length of the third side, the points are collinear, and they do not form a triangle. This condition is often referred to as forming a degenerate triangle.
How can I use the Triangle Inequality Theorem to find possible side lengths?
If you know the lengths of two sides of a triangle, you can determine the range of possible lengths for the third side by applying the inequalities. For example, if $a$ and $b$ are known, then: $| a - b | < c < a + b$ gives you the range for $c$ .
Is the Triangle Inequality Theorem applicable in coordinate geometry?
Yes, the Triangle Inequality Theorem can be applied in coordinate geometry by calculating distances between points using the distance formula and verifying the inequalities.
Can the Triangle Inequality Theorem be used in real-life situations?
Yes, it is useful in various fields such as construction, engineering and design, where determining whether certain lengths can form stable structures or shapes is essential.
Are there any specific conditions where the Triangle Inequality Theorem does not hold?
The theorem holds true for all triangles in Euclidean geometry. However, it may not apply in non-Euclidean geometries or under certain abstract mathematical conditions.
How does the Triangle Inequality Theorem relate to other geometric concepts?
The theorem is related to various other geometric principles, including similarity and congruence of triangles, and helps in understanding the properties of polygons and shapes in general.

Triangle Inequality Theorem

What is Triangle Inequality Theorem?

Solved Examples on Triangle Inequality Theorem

FAQs on Triangle Inequality Theorem

What is the Triangle Inequality Theorem?

Why is the Triangle Inequality Theorem important?

Does the Triangle Inequality Theorem apply to all triangles?

What happens if the Triangle Inequality Theorem is not satisfied?

Can the Triangle Inequality be an equality?

How can I use the Triangle Inequality Theorem to find possible side lengths?

Is the Triangle Inequality Theorem applicable in coordinate geometry?

Can the Triangle Inequality Theorem be used in real-life situations?

Are there any specific conditions where the Triangle Inequality Theorem does not hold?

How does the Triangle Inequality Theorem relate to other geometric concepts?

Triangle Exterior Angle Theorem

Triangle Similarity Theorems